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Algebra Logika, 2013, Volume 52, Number 1, Pages 57–63 (Mi al571)  

This article is cited in 16 scientific papers (total in 16 papers)

Recognizability of alternating groups by spectrum

I. B. Gorshkov

Sobolev Institute of Mathematics, Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russia

Abstract: The spectrum of a group is the set of its element orders. A finite group $G$ is said to be recognizable by spectrum if every finite group that has the same spectrum as $G$ is isomorphic to $G$. It is proved that simple alternating groups An are recognizable by spectrum, for $n\ne6,10$. This implies that every finite group whose spectrum coincides with that of a finite non-Abelian simple group has at most one non-Abelian composition factor.

Keywords: finite group, simple group, alternating group, spectrum of group, recognizability by spectrum.

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English version:
Algebra and Logic, 2013, 52:1, 41–45

Bibliographic databases:

UDC: 512.542
Received: 18.07.2012
Revised: 04.12.2012

Citation: I. B. Gorshkov, “Recognizability of alternating groups by spectrum”, Algebra Logika, 52:1 (2013), 57–63; Algebra and Logic, 52:1 (2013), 41–45

Citation in format AMSBIB
\by I.~B.~Gorshkov
\paper Recognizability of alternating groups by spectrum
\jour Algebra Logika
\yr 2013
\vol 52
\issue 1
\pages 57--63
\jour Algebra and Logic
\yr 2013
\vol 52
\issue 1
\pages 41--45

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    This publication is cited in the following articles:
    1. I. B. Gorshkov, “Recognizability of symmetric groups by spectrum”, Algebra and Logic, 53:6 (2015), 450–457  mathnet  crossref  mathscinet  isi
    2. E. Jabara, D. Lytkina, A. Mamontov, “Recognizing $M_{10}$ by spectrum in the class of all groups”, Int. J. Algebr. Comput., 24:2 (2014), 113–119  crossref  mathscinet  zmath  isi  elib  scopus
    3. Yu. V. Lytkin, “Groups critical with respect to the spectra of alternating and sporadic groups”, Siberian Math. J., 56:1 (2015), 101–106  mathnet  crossref  mathscinet  isi  elib  elib
    4. N. V. Maslova, “Finite simple groups that are not spectrum critical”, Proc. Steklov Inst. Math. (Suppl.), 292, suppl. 1 (2016), 211–215  mathnet  crossref  mathscinet  isi  elib
    5. N. V. Maslova, “Finite groups with arithmetic restrictions on maximal subgroups”, Algebra and Logic, 54:1 (2015), 65–69  mathnet  crossref  crossref  mathscinet  isi
    6. M. A. Grechkoseeva, A. V. Vasil'ev, “On the structure of finite groups isospectral to finite simple groups”, J. Group Theory, 18:5 (2015), 741–759  crossref  mathscinet  zmath  isi  elib  scopus
    7. A. V. Vasil'ev, “On finite groups isospectral to simple classical groups”, J. Algebra, 423 (2015), 318–374  crossref  mathscinet  zmath  isi  scopus
    8. A. M. Staroletov, “On recognition of alternating groups by prime graph”, Sib. elektron. matem. izv., 14 (2017), 994–1010  mathnet  crossref
    9. Yu. V. Lytkin, “On finite groups isospectral to $U_3(3)$”, Siberian Math. J., 58:4 (2017), 633–643  mathnet  crossref  crossref  isi  elib  elib
    10. Yuri V. Lytkin, “On finite groups isospectral to the simple groups $S_4(q)$”, Sib. elektron. matem. izv., 15 (2018), 570–584  mathnet  crossref
    11. M. A. Grechkoseeva, “On spectra of almost simple extensions of even-dimensional orthogonal groups”, Siberian Math. J., 59:4 (2018), 623–640  mathnet  crossref  crossref  isi  elib
    12. Yuri V. Lytkin, “On finite groups isospectral to the simple group $S_4(3)$”, Sib. elektron. matem. izv., 16 (2019), 1561–1566  mathnet  crossref
    13. Grechkoseeva M.A. Vasil'ev A.V. Zvezdina M.A., “Recognition of Symplectic and Orthogonal Groups of Small Dimensions By Spectrum”, J. Algebra. Appl., 18:12 (2019), 1950230  crossref  mathscinet  zmath  isi  scopus
    14. Gorshkov I. Staroletov A., “on Groups Having the Prime Graph as Alternating and Symmetric Groups”, Commun. Algebr., 47:9 (2019), 3905–3914  crossref  mathscinet  zmath  isi  scopus
    15. Yang N. Grechkoseeva M.A. Vasil'ev A.V., “on the Nilpotency of the Solvable Radical of a Finite Group Isospectral to a Simple Group”, J. Group Theory, 23:3 (2020), 447–470  crossref  mathscinet  zmath  isi  scopus
    16. A. M. Staroletov, “O kompozitsionnykh faktorakh konechnykh grupp, izospektralnykh prostym klassicheskim gruppam”, Sib. matem. zhurn., 62:2 (2021), 422–440  mathnet  crossref
  • Алгебра и логика Algebra and Logic
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